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231 Tutorial Sheet 512 11 November 2005 Useful facts: • To evaluate the line integral for a parameterized curve: Z Z t2 dr F · dt F · dl = dt t1 c (1) where t1 and t2 are the parameter values corresponding to the begining and end of the curve. • To evaluate the surface integral for a parameterized surface: Z Z Z Z dr dr F · dA = F· dudv × du dv S S (2) Questions 1. (a) Compute the flux of the vector field F = xi+2yj+zk out of the cylinder defined by x2 + y 2 = 1 and 0 ≤ z ≤ 1. (b) What is the flux if the vector field is replaced with F = r = xi + yj + zk? (c) Find the flux of F = z 2 k upwards through the part of the sphere x2 +y 2 +z 2 = a2 in the first octant of R3 (in the first octant all three coordinates x, y and z are positive). 2. Find the flux of F = 2xi + yj + zk upwards through the surface with parametrization r(u, v) = u2 vi + uv 2 j + v 3 k where u and v range from 0 to 1. 1 2 Conor Houghton, [email protected], see also http://www.maths.tcd.ie/~houghton/231 Including material from Chris Ford, to whom many thanks. 1